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A360146 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 10 elements.

%I A360146 #15 Mar 16 2023 17:08:53
%S A360146 210,420,630,840,1050,1260,1470,1680,1890,2100,2520,2730,2940,3150,
%T A360146 3360,3570,3780,3990,4200,4410,4830,5040,5250,5460,5670,5880,6090,
%U A360146 6300,6510,6720,7140,7350,7560,7770,7980,8190,8400,8610,8820,9030,9450,9660,9870,10080,10290,10500,10710,10920
%N A360146 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 10 elements.
%C A360146 The 10 elements are not necessarily consecutive primes.
%C A360146 All the terms are positive multiples of 210 = 7# but are not multiples of 11 and also must not belong to A206045, where the first term is 1536160080; indeed, terms d' in A206045 correspond to the longest possible APs of primes that have exactly 11 elements with these common differences d'.
%C A360146 A342309(d) gives the first element of the smallest AP with 10 elements whose common difference is a(n) = d.
%H A360146 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/3940-a1880-np-en-pa">A1880. NP en PA</a> (prime numbers in arithmetic progression) (in French).
%H A360146 Wikipedia, <a href="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression">Primes in arithmetic progression</a>.
%H A360146 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>.
%F A360146 m is a term iff A123556(m) = 10.
%e A360146 d = 210 is a term because the longest possible APs of primes with common difference d = 210 all have 10 elements. The first such AP is (199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089), then 2299 = 11*209.
%e A360146 d = 420 is another term because the longest possible APs of primes with common difference d = 420 all have 10 elements; the first such APs start with 52879, 3544939, ... The smallest one is (52879, 53299, 53719, 54139, 54559, 54979, 55399, 55819, 56239, 56659), then 57079 = 11*5189.
%o A360146 (PARI) A053669(n) = forprime(p=2, , if(n%p, return(p)));
%o A360146 f(n) = my(p=A053669(n)); for (i=1, p-1, if (!isprime(p+i*n), return(p-1))); p; \\ A123556
%o A360146 isok(n) = f(n) == 10; \\ _Michel Marcus_, Mar 10 2023
%Y A360146 Cf. A123556, A173919, A342309.
%Y A360146 Common differences for longest possible APs of primes with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), this sequence (k=10), A206045 (k=11).
%K A360146 nonn
%O A360146 1,1
%A A360146 _Bernard Schott_, Mar 09 2023